Simple Particle Needs
First question: it can be done in Blender in versions before 2.5, for example in now very old 2.49b. The file locations for download is here [1]. The mechanism you are looking for is called 'reactor particles'. It was unfortunately not re-implemented in 2.5+ series.

In Blender versions from 2.5 to 2.83 you can not do the effect of spawning particle with another particle easily. Particle system in Blender in it's current form is on it's end-of-life state, and there will be no updated to it, because it is planned to replace it by Particle Nodes. Example [2,3]. Some workaround for this poor situation is to introduce Animation Nodes [4] and find your way thorough them. This requires a lot of learning and is considered to be advanced stuff, so no point to explain it here. You really need a tutorial. 

Second question: animation of flow can with direction can be done kind of easy using small custom object arrows and particle flow controllers like 'curve field' (curve guide) or 'keyed' particles [6]. This however requires also a tutorial, for example this (first from google): [5]


https://download.blender.org/release/Blender2.49b/
https://www.youtube.com/watch?v=A05UghlpVPs
https://www.youtube.com/watch?v=2tKpHuq9qEk
https://www.youtube.com/watch?v=8pvX2_ZH1l0
https://www.youtube.com/watch?v=PYHOV3bVIG0
https://docs.blender.org/manual/en/latest/physics/particles/emitter/physics/keyed.html